Consider the set of non-zero relative integers between $-30$ and $30$; this set can be written as follows: $\{-30; -29; -28; \ldots -1; 1; \ldots; 28; 29; 30\}$. It contains 60 elements.\\
We choose from this set successively and without replacement a relative integer $a$ then a relative integer $c$.

\begin{enumerate}
  \item How many different pairs $(a; c)$ can we obtain?
\end{enumerate}

Consider the event $M$: ``the equation $ax^2 + 2x + c = 0$ has two distinct real solutions'', where $a$ and $c$ are the relative integers previously chosen.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Show that event $M$ occurs if and only if $ac < 1$.
  \item Explain why the opposite event $\bar{M}$ contains 1740 outcomes.
  \item What is the probability of event $M$? Round the result to $10^{-2}$.
\end{enumerate}